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Unit 8 · 2.4

Counting rules & factorials

Multiply the choices; repeats or not; factorials

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Rule cards

Fundamental counting principle

Steps that build ONE outcome → multiply the number of choices: e₁ × e₂ × … × eₙ.

Traps
  • Added the choices.
    10 jeans + 15 shirts + 4 shoes = 29
    ✓ 10 × 15 × 4 = 600; 4 styles × 4 stitches × 7 yarns = 112

Can items repeat?

Repeats allowed (digits, PINs) → n × n × … = nʳ. No repeats (people, cards, letters of a WORD) → n × (n − 1) × (n − 2) …

Traps
  • Counted down when repeats were allowed.
    zip codes → 10·9·8·7·6
    ✓ repeats allowed → 10⁵ = 100,000
  • Used nʳ when items can't repeat.
    no-repeat zip → 10⁵
    ✓ 10·9·8·7·6 = 30,240
  • Left 0 out of the digits.
    9 choices per digit
    ✓ digits 0–9 = 10 choices

Factorials

n! = n(n − 1)(n − 2)…1; arrange ALL n different things = n!; 0! = 1 and 1! = 1.

  1. Same n! on top and bottom? Cancel: 24!/22! = 24 × 23 = 552.
  2. 10! = 3,628,800 (exam Q22).
Traps
  • Said 0! = 0.
    0! → 0
    ✓ 0! = 1
  • Stopped one factor early or started one too high.
    6 games on a shelf → 5! = 120
    ✓ 6 games on a shelf = 6! = 720; 8 dogs = 40,320
  • Wrote nⁿ instead of n!.
    6 games → 6⁶
    ✓ 6! = 720
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